139 research outputs found
Dynamics of thermoelastic thin plates: A comparison of four theories
Four distinct theories describing the flexural motion of thermoelastic thin
plates are compared. The theories are due to Chadwick, Lagnese and Lions,
Simmonds, and Norris. Chadwick's theory requires a 3D spatial equation for the
temperature but is considered the most accurate as the others are derivable
from it by different approximations. Attention is given to the damping of
flexural waves. Analytical and quantitative comparisons indicate that the
Lagnese and Lions model with a 2D temperature equation captures the essential
features of the thermoelastic damping, but contains systematic inaccuracies.
These are attributable to the approximation for the first moment of the
temperature used in deriving the Lagnese and Lions equation. Simmonds' model
with an explicit formula for temperature in terms of plate deflection is the
simplest of all but is accurate only at low frequency, where the damping is
linearly proportional to the frequency. It is shown that the Norris model,
which is almost as simple as Simmond's, is as accurate as the more precise but
involved theory of Chadwick.Comment: 2 figures, 1 tabl
Strichartz Estimates for the Vibrating Plate Equation
We study the dispersive properties of the linear vibrating plate (LVP)
equation. Splitting it into two Schr\"odinger-type equations we show its close
relation with the Schr\"odinger equation. Then, the homogeneous Sobolev spaces
appear to be the natural setting to show Strichartz-type estimates for the LVP
equation. By showing a Kato-Ponce inequality for homogeneous Sobolev spaces we
prove the well-posedness of the Cauchy problem for the LVP equation with
time-dependent potentials. Finally, we exhibit the sharpness of our results.
This is achieved by finding a suitable solution for the stationary homogeneous
vibrating plate equation.Comment: 18 pages, 4 figures, some misprints correcte
The mixed problem in L^p for some two-dimensional Lipschitz domains
We consider the mixed problem for the Laplace operator in a class of
Lipschitz graph domains in two dimensions with Lipschitz constant at most 1.
The boundary of the domain is decomposed into two disjoint sets D and N. We
suppose the Dirichlet data, f_D has one derivative in L^p(D) of the boundary
and the Neumann data is in L^p(N). We find conditions on the domain and the
sets D and N so that there is a p_0>1 so that for p in the interval (1,p_0), we
may find a unique solution to the mixed problem and the gradient of the
solution lies in L^p
Thermodynamics of non-local materials: extra fluxes and internal powers
The most usual formulation of the Laws of Thermodynamics turns out to be
suitable for local or simple materials, while for non-local systems there are
two different ways: either modify this usual formulation by introducing
suitable extra fluxes or express the Laws of Thermodynamics in terms of
internal powers directly, as we propose in this paper. The first choice is
subject to the criticism that the vector fluxes must be introduced a posteriori
in order to obtain the compatibility with the Laws of Thermodynamics. On the
contrary, the formulation in terms of internal powers is more general, because
it is a priori defined on the basis of the constitutive equations. Besides it
allows to highlight, without ambiguity, the contribution of the internal powers
in the variation of the thermodynamic potentials. Finally, in this paper, we
consider some examples of non-local materials and derive the proper expressions
of their internal powers from the power balance laws.Comment: 16 pages, in press on Continuum Mechanics and Thermodynamic
Boundary stabilization of numerical approximations of the 1-D variable coefficients wave equation: A numerical viscosity approach
In this paper, we consider the boundary stabilization problem associated to the 1- d wave equation with both variable density and diffusion coefficients and to its finite difference semi-discretizations. It is well-known that, for the finite difference semi-discretization of the constant coefficients wave equation on uniform meshes (Tébou and Zuazua, Adv. Comput. Math. 26:337–365, 2007) or on somenon-uniform meshes (Marica and Zuazua, BCAM, 2013, preprint), the discrete decay rate fails to be uniform with respect to the mesh-size parameter. We prove that, under suitable regularity assumptions on the coefficients and after adding an appropriate artificial viscosity to the numerical scheme, the decay rate is uniform as the mesh-size tends to zero. This extends previous results in Tébou and Zuazua (Adv. Comput.Math. 26:337–365, 2007) on the constant coefficient wave equation. The methodology of proof consists in applying the classical multiplier technique at the discrete level, with a multiplier adapted to the variable coefficients
Control and stabilization of waves on 1-d networks
We present some recent results on control and stabilization of waves on 1-d networks.The fine time-evolution of solutions of wave equations on networks and, consequently, their control theoretical properties, depend in a subtle manner on the topology of the network under consideration and also on the number theoretical properties of the lengths of the strings entering in it. Therefore, the overall picture is quite complex.In this paper we summarize some of the existing results on the problem of controllability that, by classical duality arguments in control theory, can be reduced to that of observability of the adjoint uncontrolled system. The problem of observability refers to that of recovering the total energy of solutions by means of measurements made on some internal or external nodes of the network. They lead, by duality, to controllability results guaranteeing that L 2-controls located on those nodes may drive sufficiently smooth solutions to equilibrium at a final time. Most of our results in this context, obtained in collaboration with R. Dáger, refer to the problem of controlling the network from one single external node. It is, to some extent, the most complex situation since, obviously, increasing the number of controllers enhances the controllability properties of the system. Our methods of proof combine sidewise energy estimates (that in the particular case under consideration can be derived by simply applying the classical d'Alembert's formula), Fourier series representations, non-harmonic Fourier analysis, and number theoretical tools.These control results belong to the class of the so-called open-loop control systems.We then discuss the problem of closed-loop control or stabilization by feedback. We present a recent result, obtained in collaboration with J. Valein, showing that the observability results previously derived, regardless of the method of proof employed, can also be recast a posteriori in the context of stabilization, so to derive explicit decay rates (as) for the energy of smooth solutions. The decay rate depends in a very sensitive manner on the topology of the network and the number theoretical properties of the lengths of the strings entering in it.In the end of the article we also present some challenging open problems
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